Fundamentals of ThermodynamicsPart 2

Ashok Kumar Nallathambi, Prof.E.SpechtInstitute of Fluid Dynamics and Thermodynamics

Otto-von-Guericke-University, Magdeburg 39106, Germany

April 28, 2008

1 Equivalence of Kelvin-Plank and Clausius statements

Kelvin-Planks and Clausiuss statements are two parallel statements of the second law and areequivalent in all aspects. The violation of the one statement implies the violation of the second,and vice-versa.

2 Reversibility and Irreversibility

A reversible process is one which is performed in such a way that at the conclusion of the process,both the system and the surroundings may be restored to their initial states, without producingany changes in the rest of the universe. A reversible process is carried out infinitely slowly (takesinfinite time) with an infinitesimal gradient, so that every state passed through by the systemis an equilibrium state. Any natural process carried out with a finite gradient is an irreversibleprocess. All spontaneous process are irreversible. If the time allowed for a process to occur isinfinitely large, even though the gradient is finite, the process becomes reversible.

2.1 Causes of irreversibility

Lack of equilibrium during the process : The lack of mechanical, thermal and chemicalequilibrium between the system and its surroundings,or between two systems, or tow partsof the same system, causes a spontaneous change which is irreversible. eg: (a) heat transferthrough a finite temperature difference, (b) lack of pressure equilibrium within the interiorof the system or between the system and the surroundings, and (c) free expansion.

Involvement of dissipative effects: If the work done on the system without producing anequivalent increase in the kinetic or potential energy of the system and it only increasesthe molecular internal energy (U), will increase the irreversibility of the system. eg: (a)friction, (b) paddle wheel work transfer, and (c) transfer of electricity through a resistor.

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2.2 PMM3

The continual motion of a movable device in the complete absence of friction is known as per-petual motion machine of the third kind. Unlike PMM2, PMM3 violates both the Kelvin-Plankand Clausius statements.

2.3 Conditions for irreversibility

A process will be reversible when it is performed in such a way that the system is at all timesinfinitesimally near a state of thermodynamic equilibrium and in the absence of dissipative effectof any form. Reversible processes are, therefore, purely ideal, limiting cases of actual processes.

3 Carnot cycle

A Carnot cycle is an ideal hypothetical, reversible cycle in which following four reversibleprocesses occur:

1. A reversible isothermal heat addition(Q1) process

2. A reversible adiabatic work delivering (WT ) process

3. A reversible isothermal heat rejecting (Q2) process

4. A reversible adiabatic work consuming (WP ) process

From the 1st law,Q1 Q2 = WT WP (1)

= 1 Q2Q1

(2)

A reversible Carnot engine (E) can also act as a reversed heat engine (). The reversed Carnotengine is none other than the reversible heat pump of reversible refrigerator.

4 Carnots Theorem

Carnot theorem states that of all the heat engines operating between a given constant tempera-ture source and a given constant temperature sink, none has a higher efficiency than a reversibleengine.

Corollary: The efficiency of all reversible heat engines operating between the same tempera-ture levels is the same.

The efficiency of a reversible engine is independent of the nature or amount of workingsubstance undergoing the cycle.

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5 Absolute thermodynamic temperature scale

Kelvin proposed a relationQ1Q2

=T1T2

(3)

where T is known as absolute thermodynamic temperature. The Kelvin temperature scale isindependent of the peculiar characteristics of any particular substance.

6 Fowler-Guggenheim statement of Third Law of Ther-

modynamics

It is impossible by any procedure, no matter how idealized, to reduce any system to the absolutezero of temperature (0 K or -273.16oC) in a finite number of operations.

7 Efficiency of the reversible heat engine

Using Kelvin temperature relation, the efficiency of the reversible heat engine can be rewrittenas:

rev = max = 1 (Q2Q1

)rev

= 1 T2T1

(4)

Similarly, COP of the reversible refrigerator and heat pump can be given as:

[COPrefr]rev =T2

T1 T2 (5)

[COPHP]rev =T1

T1 T2 (6)

8 Types of Irreversibility

Two types of irreversibility can be given as:

1. Internal irreversibility : due to internal dissipative effects like friction, turbulence, electricalresistance, magnetic hysteresis, etc. with in the system.

2. External irreversibility : refers to the irreversibility occurring at the system boundarylike heat interaction with the surroundings due to finite temperature gradient, pressuregradient, etc.

9 Important points

Two constant property lines can never intersect each other in a thermodynamic propertyplot.

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Through one point in a thermodynamic property plot, there can pass only one reversibleadiabatic.

Any reversible path may be substituted by a reversible zigzag path, between the same endstates, consisting of a reversible adiabatic followed by a reversible isotherm and then bya reversible adiabatic, such that the heat transferred during the isothermal process is thesame as that transferred during the original process.

10 Clausius theorem

It states that the cyclic integral of dQ/T for a reversible cycle is equal to zero.R

dQ

T= 0 (7)

For a single reversible process, R

dQ

T= dS (8)

where S is the entropy which is an extensive property of a system.Definition for Entropy:

1. Simplistic: Entropy is the portion of heat that can not be converted into useful mechanicalwork.

2. Entropy is a measure of the degree of molecular disorder existing in the system.

Entropy is a point function. Therefore, entropy change in an any irreversible process can bedetermined from the reversible process. The area under the T S plot gives the heat transferin a process.

11 Clausius Inequality

Cyclic integral of dQ/T for any irreversible or reversible cycle is less less than or equal to zero.ie,

dQ

T 0 (9)

If dQT

= 0, the cycle is reversible.

If dQT

< 0, the cycle is irreversible and possible.

If dQT

> 0, the cycle is impossible because it will violate 2nd Law.

Clausius inequality for a process process can be given as

dS dQT

(10)

for a reversible process, dS = dQT

and for an irreversible process, dS > dQT

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12 Entropy principle

Entropy of an isolated system always increases, it never decrease. Entropy of an isolated system/ universe can be given as

dSuniv 0 = dSsys + dSsurr 0 (11)When the system reaches equilibrium, the entropy of the system reaches its maximum. some ofthe applications of the entropy principle can be given as

1. Heat transfer through a finite temperature difference:

Suniv = Q12T1 T2T1 T2

2. Mixing of two fluids :

Suniv = m1 c1 ln

(TfT1

)+ m2 c2 ln

(TfT2

)if m1 = m2 = m and c1 = c2 = c,

Suniv = 2 m c ln

(T1 + T2

2T1 T2

)3. Maximum work obtainable from two finite bodies at temp. T1 and T2 as

Wmax = cp

(T1

T2

)24. Maximum work obtainable from a finite body (at T ) and thermal energy reservoir (TER)

at (To) as

Wmax = cp

[(T To) To ln

(T

To

)]

13 Entropy generation in a closed system

If the system and surrounding interact through heat transfer, the entropy change of the systemcan be given as

dS = deS + diS =dQ

T+ diS (12)

where deS is the entropy increase due to external irreversibility and diS is due to internal irre-versibility and also called as internal entropy generation. But diS 0.

The second law states that, in general, any thermodynamic process is accompanied by entropygeneration.

S2 S1 entropy change

21

dQ

T entropy transfer

= Sgenentropy production

(13)

Heat transfer is always accompanied with an en entropy transfer. And also, Sgen 0.Even though entropy is a point function, entropy generation is not a point function. Entropy

generation is a path function and it is not a property of a system. The amount of entropygeneration quantifies the irreversibility of the process.

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14 Free energy

The part of the low grade energy which is available for conversion is referred as available en-ergy (exergy) and the part which, according to the second law, must be rejected, is known asunavailable energy (anergy). The maximum available energy is known as free energy.

Helmholtz free energy function: = U T SGibbs free energy function: G = H T S

15 Ideal gas equation

A hypothetical gas which obeys the law p v = R T at all pressures and temperatures is calledas an ideal gas. The universal gas constant (R) can be determined from

R =1.01325 105 N/m2 22.4 m3/Kg mol

273.15 K= 8.3143 KJ/Kg mol K (14)

The molar volume (v) can be replaced by the total volume (V ) using v = V/n, therefore

p V = n R T (15)

where n is the number of moles and it is defined as the ratio between the mass of the gas (m)and the molecular weight of the gas (). using n = m/, Eq. 15 becomes

p V = mR

T

= m R T (16)

where R is the characteristic gas constant and R = R.

The total number of molecules (N) in the gas can be given as N = n A. where A is theAvogadros number and is equal to 6.023 1026 molecules/Kg mol.

The Boltzmann constant(K) is defined as the ratio between the universal gas constant andAvogadros number.

K =R

A= 1.38 1023 J/molecule K (17)

Finally,

p V = m R T

= N K T (18)

The specific heat and gas constant and related as

cp = R

1 , cv =R

1 and =cpcv

(19)

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